# Complex numbers fundamentals

> IB Mathematics: Analysis and Approaches HL · Number & Algebra
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u1-complex-numbers-fundamentals/

This sub-topic introduces core definitions, notation, and representations of complex numbers, covering basic operations, rectangular/polar forms, and geometric interpretation, forming the foundation for all advanced complex number work.

**Prerequisites:** [Quadratic equations and discriminants](https://www.owlsprep.com/study/ib-math-aa-hl-u1-quadratic-discriminants/); Basic trigonometry and coordinate geometry

## Learning objectives

- Define the imaginary unit and core properties of complex numbers
- Convert between rectangular and polar forms of complex numbers
- Represent complex numbers on an Argand diagram
- Calculate the modulus and principal argument of any complex number
- Perform basic addition and multiplication of complex numbers in rectangular form

## Core Definitions and Rectangular Form

**Complex Number (Rectangular Form)** — A number consisting of a real part $a = \text{Re}(z)$ and imaginary part $b = \text{Im}(z)$, where $i^2 = -1$, and $a,b$ are real numbers.

*Notation:* $z = a + bi$

*Example:* For $z = 3 - 2i$, $\text{Re}(z) = 3$, $\text{Im}(z) = -2$ (not $-2i$).

Basic operations on complex numbers in rectangular form follow standard algebraic rules, with one key adjustment: replace every occurrence of $i^2$ with $-1$ when simplifying.

1. Addition: $(a+bi) + (c+di) = (a+c) + (b+d)i$, add real and imaginary parts separately
2. Subtraction: $(a+bi) - (c+di) = (a-c) + (b-d)i$, same structure as addition
3. Multiplication: Expand like a binomial product, substitute $i^2 = -1$, then group terms

**Worked example:** Given $z_1 = 4 + 2i$ and $z_2 = 1 - 3i$, calculate (a) $z_1 + z_2$, (b) $z_1 z_2$.

1. For addition, group real and imaginary components:
2. $$z_1 + z_2 = (4 + 1) + (2 + (-3))i = 5 - i$$
3. For multiplication, expand the binomial product:
4. $$(4 + 2i)(1 - 3i) = 4(1) + 4(-3i) + 2i(1) + 2i(-3i) = 4 - 12i + 2i - 6i^2$$
5. Substitute $i^2 = -1$ and simplify:
6. $$= 4 - 10i - 6(-1) = 10 - 10i$$

## Geometric Representation: The Argand Diagram

Every complex number can be represented as a point or position vector on a special coordinate plane called the Argand diagram (or complex plane).

**Argand Diagram** — A coordinate plane where the horizontal axis is the real axis (for $\text{Re}(z)$) and the vertical axis is the imaginary axis (for $\text{Im}(z)$). A complex number $z = a + bi$ maps to the point $(a, b)$.

> **info**
>
> Addition of complex numbers on the Argand diagram follows the parallelogram rule for vector addition, just like real vector addition.

**Worked example:** Plot $z_1 = 3 + i$, $z_2 = -2 - 2i$, and $z_1 + z_2$ on an Argand diagram, and confirm the addition matches vector addition.

1. First, identify coordinates for each point: $z_1 = (3,1)$, $z_2 = (-2,-2)$
2. Calculate the sum to get its coordinates:
3. $$z_1 + z_2 = (3-2) + (1-2)i = 1 - i \rightarrow (1, -1)$$
4. When plotted, the point $(1,-1)$ is the resultant of the two position vectors, which follows the parallelogram rule for vector addition.

## Modulus and Principal Argument

Polar form of a complex number is defined by two key properties: the modulus (distance from the origin) and the argument (angle from the positive real axis).

**Modulus and Principal Argument** — For $z = a + bi$, modulus $|z| = \sqrt{a^2 + b^2}$ is the distance from the origin. The principal argument $\arg(z)$ is the anticlockwise angle from the positive real axis, defined for IB in the range $-\pi < \arg(z) \leq \pi$.

*Notation:* $|z|$, $\arg(z)$

> **warning**
>
> Always check which quadrant your complex number lies in before calculating the argument. The $\arctan$ function only outputs values between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, so you need to adjust for second and third quadrants.

**Worked example:** Find the modulus and principal argument of $z = -1 + \sqrt{3}i$.

1. Calculate the modulus using Pythagoras' theorem:
2. $$|z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$$
3. Identify the quadrant: $\text{Re}(z) < 0$, $\text{Im}(z) > 0$, so $z$ lies in the second quadrant. Calculate the reference angle:
4. $$\tan \alpha = \left| \frac{\text{Im}(z)}{\text{Re}(z)} \right| = \left| \frac{\sqrt{3}}{-1} \right| = \sqrt{3} \implies \alpha = \frac{\pi}{3}$$
5. Adjust the angle to get the principal argument for the second quadrant:
6. $$\arg(z) = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$$

## Polar (Modulus-Argument) Form

Using right triangle trigonometry on the Argand diagram, we can write any non-zero complex number in polar form, which is more useful for multiplication, division, and raising to powers than rectangular form.

**Polar Form of a Complex Number** — For any non-zero complex number with modulus $r = |z|$ and principal argument $\theta = \arg(z)$, $z$ can be written as $r(\cos \theta + i \sin \theta)$, where $a = r \cos \theta$ and $b = r \sin \theta$ for $z = a + bi$.

*Notation:* $z = r(\cos \theta + i \sin \theta)$

**Worked example:** Convert $z = 1 - i$ from rectangular form to polar form.

1. First calculate the modulus $r$:
2. $$r = |z| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$$
3. Find the principal argument $\theta$: $z$ is in the fourth quadrant, so:
4. $$\tan \theta = \frac{-1}{1} = -1 \implies \theta = -\frac{\pi}{4}$$
5. Write the final polar form:
6. $$z = \sqrt{2} \left( \cos \left(-\frac{\pi}{4}\right) + i \sin \left(-\frac{\pi}{4}\right) \right)$$

## Common pitfalls

- **Wrong:** Stating $\text{Im}(z) = bi$ for $z = a + bi$, instead of just $b$.
  - Why it fails: The imaginary part is defined as the real scalar multiple of $i$, it does not include $i$ itself.
  - Correct: For $z = 5 - 3i$, $\text{Im}(z) = -3$, not $-3i$.
- **Wrong:** Writing $\arg(z) = \arctan\left(\frac{b}{a}\right)$ without adjusting for quadrant.
  - Why it fails: Arctangent only returns values between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, so it is incorrect for second and third quadrants.
  - Correct: Find the reference angle first, then adjust it to get the correct principal argument based on the quadrant of $z$.
- **Wrong:** Simplifying $\sqrt{-a} \times \sqrt{-b} = \sqrt{ab}$ for positive $a,b$.
  - Why it fails: The square root identity $\sqrt{x} \sqrt{y} = \sqrt{xy}$ only holds for non-negative real $x,y$.
  - Correct: Rewrite $\sqrt{-a} = i\sqrt{a}$ and $\sqrt{-b} = i\sqrt{b}$ first, then multiply to get $-\sqrt{ab}$.
- **Wrong:** Using the range $0 \leq \arg(z) < 2\pi$ for principal argument.
  - Why it fails: The IB AA HL syllabus requires principal argument to be in the range $-\pi < \arg(z) \leq \pi$ unless stated otherwise.
  - Correct: Subtract $2\pi$ from any angle greater than $\pi$ to get it into the principal range.
- **Wrong:** Calculating modulus as $\sqrt{a^2 - b^2}$ instead of the sum.
  - Why it fails: Confusion between Pythagoras' theorem and difference of squares.
  - Correct: Modulus is distance from the origin, so always use $|z| = \sqrt{a^2 + b^2}$.

## Cheatsheet

| Concept | Formula/Rule |
| --- | --- |
| Rectangular Form | $z = a + bi$, $\text{Re}(z)=a$, $\text{Im}(z)=b$, $i^2=-1$ |
| Modulus | $\|z\| = \sqrt{a^2 + b^2}$ |
| Principal Argument Range | $-\pi < \arg(z) \leq \pi$ |
| Addition | $(a+bi)+(c+di) = (a+c)+(b+d)i$ |
| Multiplication | $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$ |
| Polar Form | $z = r(\cos\theta + i\sin\theta)$, $r=\|z\|$, $\theta = \arg(z)$ |
| Argand Coordinates | $z=a+bi \leftrightarrow (a, b)$ |

## What's next

A strong grasp of complex number fundamentals is required for all subsequent complex number topics in IB AA HL, from multiplication of complex numbers in polar form to De Moivre's theorem and finding roots of complex numbers. It also underpins the complex conjugate root theorem for polynomials, which is a common exam question topic. Geometric intuition for complex numbers built here will help you recognize rotation and scaling properties of complex multiplication, which often appear in extended response questions.

- [Polar and exponential form of complex numbers](https://www.owlsprep.com/study/ib-math-aa-hl-u1-polar-and-exponential-form-of/)
- [De Moivre's theorem](https://www.owlsprep.com/study/ib-math-aa-hl-u1-de-moivre-s-theorem/)

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